TThermoplasticityas a nonsmooth phenomenon
Fran¸cois Demoures , September 21, 2018
Abstract
This paper develops the variational multisymplectic formulation of nonsmoothelastoplastic phenomena, where the rate of change of plastic strain and the asso-ciated thermodynamic entropy evolve by jumps. The formulation relies on convexanalysis to describe the plastic non smoothness.
Contents EPFL, Doc & Postdoc Alumni DFJC/DGEP [email protected] a r X i v : . [ m a t h . NA ] F e b Conclusion 24References 24
Plasticity.
The first important results concerning plasticity are due to Tresca[1872] and Saint-Venant [1871a,b,c]. See Maugin [2016] for an comprehensivehistorical review, and Lubliner [1990] for a general overview of the subjectThe plasticity theory, we consider in this article, was defined in Hill [1950] forthe maximal dissipation principle, Green and Naghdi [1965] for the formal addi-tive decomposition of the finite Lagrangian strain tensor, Rice [1970, 1971] for thetheoretical foundations of inelastic constitutive laws for solids, Rockafellar [1970],Moreau [1973, 1976] for the convex analysis formulation, and Suquet [1979] wherethe existence of perfect plastic solutions is investigated. The classical multiplica-tive decomposition F = F e F p is due to Bilby, Gardner, and Stroh [1957], Kr¨oner[1960], Lee and Liu [1967].In Simo and Hughes [1998], and Simo [1998] was developed an overview ofnumerical analysis dedicated to the simulation of problems involving plastic de-formation. More recent complements can be found in, e.g. Gurtin [2000], Armero[2008], Clayton and Bammann [2009].Regarding rheological thermodynamics, we refer to the following books andpapers: Landau and Lifshitz [1959], Truesdell and Noll [1965], Truesdell [1968],Ziegler and Wehrli [1987], Ottinger [2005], Gurtin, Fried, and Anand [2010] andMaugin [2011]. Concerning thermoplasticity, we refer to: Eckart [1948], Naghdi[1960], Ziegler [1963], Green and Naghdi [1966], Maugin [1992]. For numericalstudy and simulation of thermoplasticity see, for example, Simo [1998] § e refer to Mart´ınez et all [2008], Clayton, McDowell, and Bammann [2006b], Fres-sengeas, Taupin, and Capolungo [2011], Yavari and Goriely [2012]. Multisymplectic formulation.
The plasticity phenomenon will be formu-lated within the context of multisymplectic continuum mechanics (Gotay, Isen-berg, Marsden [2006]) and in particular of multisymplectic nonsmooth continuummechanics (Fetecau, Marsden, and West [2003]).
Rheological model.
The elastoplastic material exhibits both plastic and elas-tic behaviour. One can build up a model of nonsmooth elastoplasticity by com-bining a linear elastic spring and a non-smooth frictional pad. These are knownas rheological models originating from the work of Zener [1948]. See, e.g. the rhe-ological models described in Maugin [1992], Gutzow and Schmelzer [1995], Simoand Hughes [1998], and in Lion [2000].
Goals and general framework.
In this paper, firstly we develop the multi-symplectic formulation of nonsmooth elastoplastic phenomena in §
2, with a multi-plicative decomposition of the total deformation gradient F = F e F p into an elasticdeformation part F e and a plastic deformation part F p .Secondly we develop a rheological model dedicated to crystal elastoplasticityand thermoplasticity (with temperature and entropy variables added). Accordingto Simo [1998] we admit an additive decomposition of the total strain (cid:15) of thesystem into an elastic strain (cid:15) e and a plastic strain (cid:15) p due to sliding, i.e., (cid:15) = (cid:15) e + (cid:15) p . This additive decomposition is consistent with our rheological model composed ofa spring and a frictional pad where deformations are small (see Fig. 3.1).
Summary of the main results: • In § • In § • In § We establish a link between variational multisymplectic formulation of continuummechanics and elastoplasticity through internal slip of dislocation and internalfriction due to lattice displacement (translation and rotation) which are dissipativenonsmooth dynamic phenomena. .1 Moreau viewpoint and D perfect plasticity
Plastic bodies are characterized by the fact that their shape can be changed bythe application of appropriately directed external forces, and that they retain theirso-deformed shape upon removal of such forces.Let the internal stress -tensor σ ( t, s ) and (cid:74) (cid:15) p (cid:75) t a plastic strain-rate jump ,during time evolution, which occurs at position s and time t . The yield criterion f : σ (cid:55)→ R , which confines the stresses σ to lie in the elastoplastic domain, isspecified by the following inequality constraint f ( σ ) ≤ . (2.2)Thus the set of admissible stresses is defined by E σ := { σ | f ( σ ) ≤ } . (2.3)The boundary of E σ defined by f ( σ ) = 0 is called the yield surface . “The pointsat which σ is inside the yield surface ( f ( σ ) <
0) constitute the elastic domain,while those where σ is on the yield surface form the plastic domain” (Lubliner[1990]). The set E σ is supposed to be convex.Let V the set of plastic strain-rate jumps (cid:74) (cid:15) p (cid:75) t . The set of admissible stressesand the set of rate of change of plastic strain are placed in duality by a bilinear form (cid:104)· , ·(cid:105) . The plasticity law is defined by stating the maximal dissipation principle ,i.e., the values of the stress σ ∈ E σ which correspond to some (cid:74) (cid:15) p (cid:75) t ∈ V are theelements which minimize the numerical function σ (cid:55)→ − (cid:10) (cid:74) (cid:15) p (cid:75) t , σ (cid:11) .Next, we recall the convex analysis principles which allow to describe the plas-ticity. Following Moreau [1973, 1976] the stress and the strain ( σ , (cid:74) (cid:15) p (cid:75) t ) ∈ E σ × V which verify the plasticity law, i.e., the principle of maximum dissipation underthe inequality constraint f ( σ ) ≤
0, can also be defined in an equivalent way bythe variational inequality with solution σ satisfying the condition (cid:40) σ ∈ E σ ∀ σ (cid:48) ∈ E σ : (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) ≥ (cid:10) σ (cid:48) , (cid:74) (cid:15) p (cid:75) t (cid:11) , (2.4)or in the following equivalent manners ∀ σ (cid:48) ∈ R : (cid:10) σ (cid:48) − σ , (cid:74) (cid:15) p (cid:75) t (cid:11) + I E σ ( σ ) ≤ I E σ ( σ (cid:48) ) , ⇔ (cid:74) (cid:15) p (cid:75) t ∈ ∂I E σ ( σ ) , ⇔ σ ∈ ∂I ∗ E σ ( (cid:74) (cid:15) p (cid:75) t ) , (2.5) Recall that the jump (cid:74) · (cid:75) in Maugin [1992] is defined by (cid:74) V (cid:75) := V + − V − (2.1)where V − and V + are respectively referred to t − and t + . In Fetecau, Marsden, and West [2003] thedefinition of the jump is extended to the spacetime. In this development we admit the existence of a single yield criterion in order to simplify thepresentation, but generally there is a set of constraints. here I E σ is the indicator function of E σ , i.e., I E σ ( σ ) = 0 if σ ∈ E σ and I E σ ( σ ) =+ ∞ if σ / ∈ E σ . Its polar function I ∗ E σ is the support function of E σ relative to (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) , i.e., I ∗ E σ ( (cid:74) (cid:15) p (cid:75) t ) = sup σ ∈ T ( S ) (cid:8)(cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) − I E σ ( σ ) (cid:9) = sup σ ∈ E σ (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) . (2.6)As a consequence (2.4) is also equivalent to (cid:40) σ ∈ E σ (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) = I ∗ E σ ( (cid:74) (cid:15) p (cid:75) t ) . (2.7)That is, the values of σ associated with a given (cid:74) (cid:15) p (cid:75) t ∈ V , by the plasticity law,are the elements of E σ for which the dissipated value is exactly equal to I ∗ E σ ( (cid:74) (cid:15) p (cid:75) t ).In Moreau [1973], I ∗ E σ is denoted the dissipation function . From now on we notethe plastic dissipation by D p ( σ , (cid:74) (cid:15) p (cid:75) t ) := I ∗ E σ ( (cid:74) (cid:15) p (cid:75) t ) . (2.8) Remark 2.1
We recall that ∂I E σ ( σ ) = N E σ ( σ ) is the normal cone to E σ at σ and ∂I ∗ E σ ( (cid:74) (cid:15) p (cid:75) t ) = N ∗ E σ ( (cid:74) (cid:15) p (cid:75) t ) is its dual; see Rockafellar and Wets [1998]. Moreoverwe recall that λ ∇ σ f ( σ ) ∈ N E σ ( σ ), with the Lagrange multiplier λ which satisfies λ = 0 when f ( σ ) < λ ≥ f ( σ ) = 0. Note that λ obeys to the Kuhn-Tucker complementarity conditions : λ ≥ f ( σ ) ≤ λf ( σ ) = 0 . (cid:3) Remark 2.2
An essential feature of the convex analysis principles is the impos-sibility to define (cid:74) (cid:15) p (cid:75) t as a single valued fonction of σ , nor σ as a single valuedfunction of (cid:74) (cid:15) p (cid:75) t . Indeed for (cid:74) (cid:15) p (cid:75) t = 0 corresponds for σ all the values of int( E σ ),and for σ ∈ ∂ E σ corresponds for (cid:74) (cid:15) p (cid:75) t all the values of the normal cone N E σ ( σ ).Later on, we will tackle this issue. (cid:3) Remark 2.3
Concerning plasticity, it is important to note that the constraint(2.2) is applied to the stress. There are no direct constraints on the configurationof the body. (cid:3)
The vector measure plays an important role in the subsequent development, so weshall devote this section to recall the following results.
Vector measure.
Let a Banach space X , a locally compact domain T , and thevector space H ( T ) of real continuous functions ψ : T → R with compact support.A vector measure on T with value in X , in the sense of Bourbaki [1959], is thelinear application m : H ( T ) → X such that for compact subdomain K of T , therestriction of m to H ( T, K ) is continuous for the topology of uniform convergence.If ψ ∈ H ( T ), the vector measure is noted (cid:82) ψ dm instead of m ( ψ ). See in Rockafellar and Wets [1998] for a general development of the concept of support function . ollowing Moreau [1988b], instead of a locally compact domain T , for simplifi-cation, we consider a real interval I and we admit that X is a Banach space, withmetric denoted d. Let f : I → X which is said to be of locally bounded variations on [ a, b ] iff var( f, [ a, b ]) < + ∞ ; notation f ∈ lbv([ a, b ] , X ). Where the variationvar( f, [ a, b ]) of f on [ a, b ] is defined as followsvar( f, [ a, b ]) = sup n (cid:88) i =1 d (cid:0) f ( τ i − ) , f ( τ i ) (cid:1) , with τ = a, ..., τ n = b. In this definition the supremum is taken over all strictly increasing finite sequence S : τ < τ < ... < τ n of points of [ a, b ].From Moreau [1988b] we recall the following results. Proposition 2.4
Let f ∈ lbv( I, X ) ; for every ψ ∈ H ( I ) and every θ iS ∈ [ τ i − , τ i ] ,the mapping S (cid:55)→ (cid:80) ni =1 ψ ( θ iS )( f ( τ i ) − f ( τ i − )) converges to a limit independent of θ , denoted (cid:82) ψdf . The convergence is uniform with regard to the choice of θ . Note that the linear mapping H ( I ) (cid:51) ψ (cid:55)→ (cid:82) ψdf ∈ X constitues a vectormeasure on I in the sense of Bourbaki. Where df is called the differential measure(or Stieltjes measure) of f ∈ lbv( I, X ). Radon-Nikodym property.
The Banach space X has the Radon-Nikodymproperty if, for every absolutely continuous f : I → X , the differential measure df admits a density f (cid:48) t ∈ L ( I, dt ; X ) relative to Lebesgue’s measure dt ; notation df = f (cid:48) t dt . Where L ( I, dt ; X ) is the notation for the set of X -valued functionswhich are µ -integrable (in the sense of Bourbaki) over every compact subset of I . In particular, the finite dimensional Banach space has the Radon-Nikodymproperty.
An important result, when f ∈ lbv( I, X ), deals with the df -measure of thesingleton { a } . That is, for every a ∈ I we have (cid:90) { a } f (cid:48) t dt = f + ( a ) − f − ( a ) . (2.9)Thus, we deduce a relationship between (2.9) and (2.1) when the jump is locallybounded. Remark 2.5 If X has finite dimension, any X -valued measure f is majorable .That is, there exists a nonnegative real measure µ on I such that, for every ψ ∈H ( I ) one has (cid:107) (cid:82) ψ df (cid:107) ≤ (cid:82) | ψ | dµ .Then it can be proved that, if X is a finite dimensional Banach space, every X -valued vector measure f possesses a density f (cid:48) µ ∈ L ∞ ( I, µ ; X ) relative to itsmodulus measure µ = | f | . (cid:3) Another important result is the following
Proposition 2.6 If X possesses the Radom-Nikodym property and if f ∈ lbv( I, X ) ,at Lebesgue almost every point τ of I , the function f possesses a derivative function ˙ f ( τ ) and after arbitrary extension to the whole of I , it constitutes a representativeof the density f (cid:48) t ∈ L ( I, dt ; X ) . rom this Proposition and (2.9) we establish the link between the plastic strain-rate jump and a time derivative. Thus we get dimensionally consistent results aswe will see. Remark 2.7
A natural generalization of the previous recapitulation dedicatedto the vector measures consists in replacing the Lebesgue measure dt by someprescribed nonnegative real measure µ on the interval I . (cid:3) Integral with respect to the vector measure.
Let the dual Banach space X (cid:48) of X with duality pairing (cid:104)· , ·(cid:105) . Given a vector measure f on I with value in X . For all x (cid:48) ∈ X (cid:48) , the linear mapping H ( I ) (cid:51) φ (cid:55)→ (cid:10) x (cid:48) , f ( φ ) (cid:11) = (cid:28) x (cid:48) , (cid:90) φ df (cid:29) = (cid:90) φ d ( x (cid:48) ◦ f ) ∈ R (2.10)is a real measure which depends linearly from x (cid:48) .If df = f (cid:48) µ dµ , where m (cid:48) µ is the density with respect to the positive measure µ on I , the integral of φ with respect to the vector measure f is defined by x (cid:48) ◦ f ( φ ) = (cid:90) φ (cid:10) x (cid:48) , f (cid:48) µ (cid:11) dµ, (2.11)where x (cid:48) ◦ f = (cid:10) x (cid:48) , f (cid:48) µ (cid:11) µ is a scalar measure for all x (cid:48) ∈ X (cid:48) , see Bourbaki [1959].Later on, we will make some connections between the scalar mesure (2.11) andthe dissipation (2.8) previously defined. Nonsmooth mechanics.
The fundamental theorem 2.8 which describe thevariational multisymplectic formulation of nonsmooth continuum mechanics waspresented in Fetecau, Marsden, and West [2003]. The general framework in whichthis theory was defined is field theory. The physical fields ϕ : X → Y are thesections of the covariant configuration bundle π XY : Y → X , where X is the spacetime domain with coordinates { x = t, x , ..., x n } and Y = X × M is the configuration bundle with ambient space M and coordinates { ϕ , ..., ϕ N } on it. Sothe coordinates on Y are written as ( x µ , ϕ A ) with µ = 0 , ..., n and A = 1 , ..., N .The Lagrangian density is of the form L ( x µ , ϕ A , ˙ ϕ A , ϕ A,i ) = L ( x µ , ϕ A , ˙ ϕ A , ϕ A,i ) d n +1 x, (2.12)where ˙ ϕ := ∂ϕ/∂t is the time derivative, ϕ ,i := ∂ϕ/∂x i , i = 1 , ..., n are the partialspace derivative, and d n +1 x = dx ∧ ... ∧ dx n ∧ dx . The associated action functionalis defined to be S ns ( ϕ ) := (cid:90) U X L ( x µ , ϕ A , ˙ ϕ A , ϕ A,i ) . (2.13)Stationarity of the action S ns with respect to variations δϕ yields the Euler-Lagrange field equations or covariant Euler-Lagrange (CEL) equations ∂∂t ∂L∂ ˙ ϕ A + ∂∂x i ∂L∂ϕ A,i − ∂L∂ϕ A = 0 . (2.14) e introduce (see Figure below) a manifold U with smooth closed boundary,a map φ : U → Y taken to be smooth, the diffeomorphism φ X : U → U X ⊂ X ,and a submanifold D ⊂ U , called the singularity submanifold , across which theLagrangian L may have singularities. Given D X := φ X ( D ), it is assumed thatthe singularity submanifold D X separates the interior of U X in two disjoint opensubsets U + X and U − X . The Lagrangian L is assumed to be smooth only on U X \ D X . Y := X × M π (cid:15) (cid:15) U φ (cid:57) (cid:57) φ X (cid:47) (cid:47) U X ⊂ X ϕ (cid:79) (cid:79) From the variational principle, as proved in Fetecau, Marsden, and West [2003],we derive directly the equations of motion and the jump conditions , staying onthe Lagrangian side. In particular the jump conditions are due to the differentorientations of D X when Stokes’ theorem is involved in the integration by parts. Theorem 2.8
Given a Lagrangian density L ( x µ , ϕ A , ˙ ϕ A , ϕ A,i ) , which is smoothaway from the discontinuity in D X , there exists unique derivative of the action d S ns ( ϕ ) such that for any V = ( V µ , V A ) ∈ T ϕ C compactly supported in U andany open subset U X such that U X ∩ ∂X = ∅ , d S ns ( φ ) · ( V ) = (cid:90) U + X ∪ U − X (cid:18) ∂L∂ϕ A − ∂∂t ∂L∂ ˙ ϕ A − ∂∂x i ∂L∂ϕ A,i (cid:19) · V A d n +1 x (2.15)+ (cid:90) U + X ∪ U − X (cid:18) ∂L∂x ν + ddx µ (cid:18) ∂L∂ϕ A,µ ϕ A,ν (cid:19) − dLdx ν (cid:19) V ν d n +1 x (2.16)+ (cid:90) ∂U X \ D X (cid:18) ∂L∂ϕ A,µ · V A d n x µ (cid:19) (2.17)+ (cid:90) ∂U X \ D X (cid:18)(cid:18) Lδ µν − ∂L∂ϕ A,µ ϕ A,ν (cid:19) V ν d n x µ (cid:19) (2.18)+ (cid:90) D X (cid:115) ∂L∂ϕ A,µ · V A d n x µ (cid:123) (2.19)+ (cid:90) D X (cid:115) (cid:18) Lδ µν − ∂L∂ϕ A,µ ϕ A,ν (cid:19) V ν d n x µ (cid:123) . (2.20)Where (2.15) gives the CEL equations. The time component of (2.16) is the energy-evolution equation , while its full expression formulates the balance of con-figurational forces . The last two expressions (2.19) and (2.20) are respectively the vertical jump conditions involving momenta and the horizontal jump conditions ,i.e., energy jump conditions, which are the consequence of local nonsmoothnesswhen x ∈ D X . Remark 2.9
Let an application U X (cid:51) x (cid:55)→ V ( x ) ∈ R N with locally boundedvariations on U X , see in § Where U X is the closure of U X . iscontinuities but, at every point x , the left-limit and the right-limit exist. Thenone can associate an R N -valued measure. If one has V − ( x ) (cid:54) = V + ( x ) for x ∈ D X ,where V − and V + are respectively referred to U − X and U + X , as explained in Moreau[1988a] the R N -valued measure possesses one atom x ∈ D X with jump value (cid:74) V (cid:75) ( x ) = V + ( x ) − V − ( x ).Also the jump can be interpreted as the value of a vector measure , respectively covector measure , on the set D X of the atoms.Let the case, where we have a form α . In view of the Radon-Nikodym property,every covector measure (cid:74) α (cid:75) ( x ) on the locally compact subset D X of U X ⊂ R n +1 may be represented as follows: there exists a positive scalar measure dt on D X and a density function denoted (cid:74) α (cid:75) (cid:48) t relatively to the Lebesgue measure, such thatone writes j ∗ (cid:74) α (cid:75) ( x ) = (cid:74) α (cid:75) (cid:48) t ( x ) dv D X , (2.21)where j : D X (cid:44) → U X is the inclusion and dv D X is the measure associated to thevolume form on D X .However even though we know from the Radon-Nikodym theorem that thereexists a density function, the theorem does not indicate how to calculate thisdensity function. This question will be solved through convex analysis. (cid:3) Nonsmooth continuum mechanics and inequality constraints.
Thevariational inequalities and the problems of constrained minimization were widelystudied by Moreau and Rockafellar, see e.g., Rockafellar and Wets [1998]. Fromthis viewpoint there have been developments that bear on the multisymplectic for-mulations of nonsmooth continuum mechanics when the configuration is subjectedto inequality constraints, see e.g., Fetecau, Marsden, Ortiz, and West [2003] andDemoures, Gay-Balmaz, and Ratiu [2016].For example, for frictionless contacts, the force of constraint is normal to theconcerned bodies. Following Moreau [1988a], the dt -measurable vector field (cid:74) α (cid:75) (cid:48) t satisfies (cid:74) α (cid:75) (cid:48) t ( x ) ∈ ∂I C ( ϕ ( x )) , (2.22)where ∂I C ( ϕ ( x )) is the normal cone to the admissible contact domain C ⊂ M in ϕ ( x ) and the density function denoted (cid:115) ∂L∂ϕ A,ν ( x ) N ν ( x ) (cid:123) (cid:48) t is defined to be the vector field associated to the vertical jump (2.19), where N ν is the normal vector to D X .From (2.22) and the properties of a normal cones we can deduce that thereexists a Lagrange multiplier λ such that we get for every V A (cid:90) D X (cid:115) ∂L∂ϕ A,ν ( x ) N ν ( x ) (cid:123) (cid:48) t · V A ( ϕ ( x )) dv D X = (cid:90) D X λ i ( x ) ψ i,A ( ϕ ( x )) · V A ( ϕ ( x )) dv D X . (2.23) We recall that the vectors in the normal cone ∂I C ( ϕ ( x )) are of the form λ ( x ) · ∇ ψ ( ϕ ( x )) where λ are the Lagrange multipliers and ψ are the inequalities constraints ψ ( ϕ ) ≤ f we consider the time component relative to the Lebesgue’s measure dt , due toProposition 2.6, we can deduce that there exists a time derivative which constitutesa representative of the momenta jump. As a confirmation, the equation (2.23) isdimensionally consistent, i.e., on the left we have the time derivative of a momentaand a force on the right. Concerning the left-hand side of (2.23), one has (cid:90) D X (cid:115) ∂L∂ϕ A,ν ( x ) N ν ( x ) (cid:123) (cid:48) t · V A ( ϕ ( x )) dv D X = (cid:90) D X (cid:115) ∂L∂ϕ A,ν ( x ) N ν ( x ) · V A ( ϕ ( x )) (cid:123) (cid:48) t dv D X , where we used the continuity of the vector field V . Also we get the statementestablished in Demoures, Gay-Balmaz, and Ratiu [2016]. We refer to Moreau [1973, 1976, 1982, 1988a,b] in addition to Fetecau, Marsden,and West [2003] for this development.In order to associate the variational multisymplectic formulation and convexanalysis, we will take into account the variational inequalities (2.4) where con-straints (2.2) are included. Additionally, we will use the properties of the varia-tional inequalities which can be expressed in different equivalent ways, in particularthrough the plastic dissipation.
The material frame indifference states that if we viewthe configuration from a rotated point of view, then the stress transforms bythe same rotation. Also, if we want that the stored energy function W ( x, ϕ, F e )satisfies this property it must depends on the elastic gradient deformation F e through right Cauchy-Green deformation tensor C e = F Te F e (see Marsden andHughes [1994]).We admit that the deformation gradient takes the form of a local multiplicativedecomposition into elastic and plastic matrices as F = F e F p .The Lagrangian density is defined as follows L ( x, ϕ, ˙ ϕ, F e ) = 12 ρ ( x ) (cid:104) ˙ ϕ, ˙ ϕ (cid:105) dv ( x ) − ρ ( x ) W ( x, ϕ, C e ( F e )) dv ( x ) , (2.24)with the mass density ρ and the elastic component F e = FF − p of the deformationgradient. The components of F are F Ai = ϕ A,i . Note that, for simplicity, weconsider the Euclidean case.
Plastic dissipation through viscous regularization.
The problem toresolve is to calculate the plastic strain-rate jump and the plastic dissipation,while the material frame indifference is required. The difficulty is precisely thatthe plasticity is a nonsmooth phenomena, see § F p evolve by jumps.To solve this difficulty we recall from Moreau [1973] that we can take intoaccount at the same time several resistance laws, like viscosity and plasticity. Theintroduction of viscosity produces a regularization effect of the plastic strain called Moreau-Yosida regularization . Conversely, the plasticity can be seen as the limit f the viscoplasticity when viscosity disappears. Indeed, adding a tiny viscosityeffect to a plasticity law produce a penalty function, where the size of the penaltycoefficient is inversely proportional to the value of the viscosity coefficient η . But,when η →
0, the penalty function becomes the indicator function. So we get againa plastic law and a strain-rate which evolves by jumps.The viscoelastoplasticity is characterized by − S : D p + η q ( d ( S )), which corre-spond to the viscous regularization of the plastic dissipation S : D p through theviscous dissipation η q ( d ( S )). Where S is the symmetric second Piola-Kirchhoffstress tensor, D p is the plastic strain-rate, q is a quadratic form, and d ( S ) is thedistance between S and the convex admissible elastoplastic domain E S , character-ized by the yield conditions f ( S ) ≤
0, i.e., E S := { S | f ( S ) ≤ } . (2.25)Thus the Moreau-Yosida regularization occurs when f ( S ) ≥
0. However, as before,we have an elastic phenomena when f ( S ) < D p through the derivative ˙ F p (see in Simo [1998] for details) D p := (cid:16) C e ˙ F p F − p (cid:17) sym (2.26)where C e and ˙ F p F − p are unaffected by rigid motions superposed on the currentplacement.In order to keep only the plastic law, we remove the viscosity ( η = 0). Hencethe elastoplastic strain-rate jump, denoted (cid:74) D p (cid:75) t , and the elastoplastic dissipation D p are defined as the following limitslim η → (cid:16) C e ˙ F p F − p (cid:17) sym =: (cid:74) D p (cid:75) t , lim η → S : D p = S : (cid:74) D p (cid:75) t = D p , (2.27)where the values of C e is given at time t − . From (2.5) we recall that (cid:74) D p (cid:75) t ∈ ∂ I E S ( S ) . (2.28)Note that the rate of plastic deformation (cid:74) D p (cid:75) t is different from zero if andonly if f ( S ) = 0, i.e., the plastic components F p are preserved when f ( S ) (cid:54) = 0. Inaddition, we recall that the normal cone ∂ I E S ( S ) in S can be written in the form λ ∂ S f ( S ), where λ are Lagrange multipliers. Remark 2.10
It is important to note that the nonsmoothness of the plastic gra-dient of deformation F p , which evolves by jumps, induces nonsmoothness of thedensity Lagrangian (2.24). So, in the following, we can take into account of theTheorem 2.8. (cid:3) Remark 2.11
The introduction of viscosity in elastoplasticity and the Moreau-Yosida regularization are studied more in details in Demoures [2018b]. (cid:3)
We know that outside of the nonsmooth plastic phenomena the elastic deforma-tion are described by the CEL equations (2.14). Also we will investigate in themultisymplectic framework the vertical and horizontal jumps. ingularity submanifold. The submanifold D X ⊂ U X is the singularity sub-manifold which matches with the plastic domain ∂ E S defined by f ( S ( ϕ )) = 0.That is D X is the space-time domain locally compact such that the time evolutionof S ( ϕ ) is nonsmooth and the dissipation function D p , defined in (2.27), verifiesthe maximal dissipation principle under inequality constraint f ( S ) ≤
0. We candeduce that exists a Lagrangian multipliar λ such that D p ascertains the maximaldissipation when (cid:74) D p (cid:75) t = λ ∂ S f ( S ). Vertical variations ( V A (cid:54) = 0 ). The plasticity is due to slips and frictionsinside the body, but without impenetrability constraints on the fields ϕ , contraryto what is happening in contact mechanics, see, e.g., (2.23). So we get from (2.23)the following jumps conditions without reaction forces (contact impulsion) (cid:90) D X (cid:115) ∂L∂ϕ A,ν ( x ) N ν ( x ) V A (cid:123) (cid:48) t dv D X = 0 . (2.29) Horizontal variations ( V µ (cid:54) = 0 ). Given the fact that the set of plastic de-formation 2-tensor is a finite dimensional Banach space with the Radon-Nikodymproperty, the rate of plastic deformation (cid:74) D p (cid:75) t is the value of a vector measurewith bounded variations which possesses a density denoted (cid:74) D p (cid:75) (cid:48) t relative to ameasure dt , see in § S : (cid:74) D p (cid:75) t is a pairing betweenthe stress and the rate of plastic deformation. Such that we get a scalar measure (cid:0) S : (cid:74) D p (cid:75) (cid:48) t (cid:1) dt for all S (see (2.11)). In addition, from Proposition 2.6 we deducethat the units are respected, i.e., the elastoplastic dissipation D p , as defined in(2.27), is a power.Concerning the time component of the horizontal jump (2.20), i.e., V (cid:54) = 0, V j = 0 for j = 1 , ..., n . This is the value of a real measure with bounded variationswhich possesses a density denoted (cid:114) L − ∂L∂ϕ A,µ ˙ ϕ A (cid:122) (cid:48) t relative to a measure dt , whichcan be represented by a time derivative. As a consequence the global energy jumpcondition is given with the correct units, by (cid:90) D X (cid:115) L − ∂L∂ϕ A,µ ˙ ϕ A (cid:123) (cid:48) t V dv D X − (cid:90) D X (cid:0) S : (cid:74) D p (cid:75) (cid:48) t (cid:1) V dv D X (cid:51) , (2.30)which characterizes the intersection between the horizontal jump condition due tononsmoothness and the maximum dissipation principle.The previous results leads to the following theorem which describes the elasto-plastic behavior through the variational multisymplectic formulation. Theorem 2.12
Consider a Lagrangian density L ( x µ , ϕ A , ˙ ϕ A , F Ai ) which is smoothaway from the discontinuity D X , where deformation gradient F = F e F p is seenas the composition of elastic and plastic deformations. We assume the same hy-potheses as above on π XY : Y → X . Then φ = ( φ X , ϕ ) is a critical point of S ns relative to the constraint (2.25) on the second Piola-Kirchhoff stress tensor S ifand only if • Away from the singularity, the field φ satisfies the covariant Euler-Lagrangeequations on U X \ D X , ∂L∂ϕ A − ∂∂t ∂L∂ ˙ ϕ A − ∂∂x i ∂L∂ϕ A,i = 0 , (2.31) ogether with the balance of energy on U X \ D X . • When x ∈ D X the field φ verify the following conditions: (a) the vertical jump condition: (cid:90) D X (cid:115) ∂L∂ϕ A,ν ( x ) N ν ( x ) V A (cid:123) (cid:48) t dv D X = 0 . (2.32) (b) the global energy jump condition (time component): for all vector fields V we have (cid:90) D X (cid:115) L − ∂L∂ϕ A,µ ˙ ϕ A (cid:123) (cid:48) t V dv D X = (cid:90) D X (cid:0) S : (cid:74) D p (cid:75) (cid:48) t (cid:1) V dv D X , (2.33) where (cid:74) D p (cid:75) t = λ ∂ S f ( S ) ∈ ∂I E P ( S ) with the Lagrange multiplier λ . • On the boundary ∂U X \ D X , the field φ verifies the following conditions: (c) we have ∂L∂ϕ A,ν V A N ν = 0 , (2.34) (d) for all ν = 0 , ..., n we have (cid:18) Lδ µν − ∂L∂ϕ A,µ ϕ A,ν (cid:19) N µ = 0 . (2.35) Remark 2.13 “The fact that the constraints involve only spatial and not timederivatives means that imposing the constraints is equivalent to restricting theinfinite-dimensional configuration manifold used to formulate the theory as a tra-ditional Hamiltonian or Lagrangian field theory. In this case, the constraint issimply a holonomic or configuration constraint and it is known that restrictingHamiltons principle to the constraint submanifold gives the correct equations forthe system. ” Marsden, Pekarsky, Shkoller, and West [2001] (cid:3)
Noether theorem.
Consider a one-parameter family φ (cid:15) of deformation map-pings that are a symmetry of the mechanical system. That is the Lagrangian L is equivariant with respect to the symmetry group action G . This impliesthe preservation of the action functional S ns ( φ ) = (cid:82) U X L ( x, ϕ A , ˙ ϕ A , F Ai ), i.e., S ns ( φ ) = S ns ( φ (cid:15) ) where φ = φ . When the plastic dissipation occurs we candeduce the integral form of Noether’s theorem, from Demoures, Gay-Balmaz, andRatiu [2016] in § U (cid:48) ⊂ U with piecewise smooth boundary and for all ξ in the Lie algebra g of the Lie group G , we have (cid:90) φ X ( D (cid:48) ) (cid:115) ∂L∂ϕ A,µ ξ AY d n x µ (cid:123) + (cid:90) ∂φ X ( U (cid:48) ) \ φ X ( D (cid:48) ) (cid:18) ∂L∂ϕ A,µ ξ AY d n x µ (cid:19) = 0 , (2.36)where ξ Y is the infinitesimal generator. Remark 2.14
Generally nonsmoothness is associated with boundary contact,and friction, and/or with interior plasticity. In such cases the question of conser-vation of symmetries must be studied under combination of different perspectives. (cid:3) ultisymplectic form formula. Concerning the Cartan form (or multi-symplectic form) and the multisymplectic form formula this will be studied in itsown for various types of nonsmooth problems in a paper to come.
The rheological model we consider was described in Simo and Hughes [1998].
The 1D rheological model (see Fig 3.1) is composed of an elastic spring of length (cid:96) at rest, with Young modulus E , and elastic strain (cid:15) e . At one end of the springwe fix a mass m , and at the opposite we fix a frictional pad which induces thefrictional strain (cid:15) p referred to as the plastic strain. Let a 1D model (see Fig 2.1) composed of an elastic spring of length ` at rest and withYoung modulus E . At one extremity of the spring we fixe a mass m , and on the otherside we fixe a frictional device characterize by its cone of friction, that is the stress maytake any value from zero up to Y . and introduce the fundamental concept of return mapping or catching up algo-rithm. As shown in Chapter this notion has a straightforward generalization tothree-dimensional models and constitutes the single most important concept incomputational plasticity. In Section we illustrate the role of these integrativealgorithms by considering the simplest finite-element formulation of the elasto-plastic boundary-value problem. We discuss the incremental form of this problemand introduce the important notion of consistent or algorithmic tangent modulus .Finally, Section generalizes the preceding ideas to accommodate rate-dependent response within the framework of classical viscoplasticity. We examinetwo possible formulations of this class of models and discuss their numerical imple-mentation. In particular, emphasis is placed on the significance of viscoplasticityas a regularization of rate-independent plasticity. This interpretation is importantin the solution of boundary-value problems where hyperbolicity of the equations inthe presence of softening can always be attained by suitable choice of the relaxationtime.For further reading on the physical background, and generalizations, seeLemaitre and Chaboche [1990]. To motivate the mathematical structure of classical rate-independent plasticity,developed in subsequent sections, we examine the mechanical response of theone-dimensional frictional device illustrated in Figure .We assume that the device initially possesses unit length (and unit area) andconsists of a spring, with elastic constant E , and a Coulomb friction element, withconstant σ Y >
0, arranged as shown in Figure . We let σ be the applied stress(force) and ε the total strain (change in length) in the device. Inspection of Figure leads immediately to the following observations: ! ! Y ! E Figure 1.1.
One-dimensional frictional device illustrating rate–independent plasticity.
Figure 2.1:
Elastoplastic bar.
We admit an additive decomposition of the total displacement x into elastic displace-ment x e due to spring and frictional displacement x p x = x e + x p . The value of the stress acting on the spring is = E✏ e = E ( ✏ ✏ p ) = E ✓ @u@s ✏ p ◆ Let the non negative variable ↵ : [0 , T ] ! denoted the internal hardening variable.We assume that we have the two following evolutionary equations˙ ↵ = | ˙ ✏ p | , and ˙ ✏ p = sign( ) , where > . (2.1)Then a yield criterion f ( , ↵ ), as a convex function, is defined f ( , ↵ ) := | | ( Y + K↵ ) . (2.2)Observe that ˙ ✏ p = @ f ( , ↵ ) can be related to convex analysis, see Moreau [1973]. Thecriterion (2.5) allows to constraint the variables , ↵ to stay in the convex elastoplasticdomain , defined as follows:= ( , ↵ ) s.t. | | ( Y + K↵ ) . (2.3)2 Multisymplectic form formula.
In Demoures, Gay-Balmaz, and Ratiu [2016] § ' . The rheological model we consider was described in Simo and Hughes [1998].
The 1D rheological model (see Fig 3.1) is composed of an elastic spring of length ` at rest, with Young modulus E , and strain ✏ e due to the elastic strain. At oneextremity of the spring we fix a mass m , and on the other side we fix a frictionaldevice which induces the frictional strain ✏ p referred to as the plastic strain. Let a 1D model (see Fig 2.1) composed of an elastic spring of length at rest and withYoung modulus E . At one extremity of the spring we fixe a mass m , and on the otherside we fixe a frictional device characterize by its cone of friction, that is the stress maytake any value from zero up to Y . and introduce the fundamental concept of return mapping or catching up algo-rithm. As shown in Chapter this notion has a straightforward generalization tothree-dimensional models and constitutes the single most important concept incomputational plasticity. In Section we illustrate the role of these integrativealgorithms by considering the simplest finite-element formulation of the elasto-plastic boundary-value problem. We discuss the incremental form of this problemand introduce the important notion of consistent or algorithmic tangent modulus .Finally, Section generalizes the preceding ideas to accommodate rate-dependent response within the framework of classical viscoplasticity. We examinetwo possible formulations of this class of models and discuss their numerical imple-mentation. In particular, emphasis is placed on the significance of viscoplasticityas a regularization of rate-independent plasticity. This interpretation is importantin the solution of boundary-value problems where hyperbolicity of the equations inthe presence of softening can always be attained by suitable choice of the relaxationtime.For further reading on the physical background, and generalizations, seeLemaitre and Chaboche [1990]. To motivate the mathematical structure of classical rate-independent plasticity,developed in subsequent sections, we examine the mechanical response of theone-dimensional frictional device illustrated in Figure .We assume that the device initially possesses unit length (and unit area) andconsists of a spring, with elastic constant E , and a Coulomb friction element, withconstant σ Y >
0, arranged as shown in Figure . We let σ be the applied stress(force) and ε the total strain (change in length) in the device. Inspection of Figure leads immediately to the following observations: ! ! Y ! E Figure 1.1.
One-dimensional frictional device illustrating rate–independent plasticity.
Figure 2.1:
Elastoplastic bar.
We admit an additive decomposition of the total displacement x into elastic displace-ment x e due to spring and frictional displacement x p x = x e + x p . The value of the stress acting on the spring is = E e = E ( p ) = E ✓ u s p ◆ Let the non negative variable : [0 , T ] ! denoted the internal hardening variable.We assume that we have the two following evolutionary equations˙ = | ˙ p | , and ˙ p = sign( ) , where > . (2.1)Then a yield criterion f ( , ), as a convex function, is defined f ( , ) := | | ( Y + K ) . (2.2)Observe that ˙ p = f ( , ) can be related to convex analysis, see Moreau [1973]. Thecriterion (2.5) allows to constraint the variables , to stay in the convex elastoplasticdomain , defined as follows:= ( , ) s.t. | | ( Y + K ) . (2.3)2 Let a 1D model (see Fig 2.1) composed of an elastic spring of length at rest and withYoung modulus E . At one extremity of the spring we fixe a mass m , and on the otherside we fixe a frictional device characterize by its cone of friction, that is the stress maytake any value from zero up to Y . and introduce the fundamental concept of return mapping or catching up algo-rithm. As shown in Chapter this notion has a straightforward generalization tothree-dimensional models and constitutes the single most important concept incomputational plasticity. In Section we illustrate the role of these integrativealgorithms by considering the simplest finite-element formulation of the elasto-plastic boundary-value problem. We discuss the incremental form of this problemand introduce the important notion of consistent or algorithmic tangent modulus .Finally, Section generalizes the preceding ideas to accommodate rate-dependent response within the framework of classical viscoplasticity. We examinetwopossibleformulationsofthisclassofmodelsanddiscusstheirnumericalimple-mentation. In particular, emphasis is placed on the significance of viscoplasticityas a regularization of rate-independent plasticity. This interpretation is importantin the solution of boundary-value problems where hyperbolicity of the equations inthe presence of softening can always be attained by suitable choice of the relaxationtime.For further reading on the physical background, and generalizations, seeLemaitre and Chaboche [1990]. To motivate the mathematical structure of classical rate-independent plasticity,developed in subsequent sections, we examine the mechanical response of theone-dimensional frictional device illustrated in Figure .We assume that the device initially possesses unit length (and unit area) andconsists of a spring, with elastic constant E , and a Coulomb friction element, withconstant σ Y >
0, arranged as shown in Figure . We let σ be the applied stress(force) and ε the total strain (change in length) in the device. Inspection of Figure leads immediately to the following observations: ! ! Y ! E Figure 1.1.
One-dimensional frictional device illustrating rate–independent plasticity.
Figure 2.1:
Elastoplastic bar.
We admit an additive decomposition of the total displacement x into elastic displace-ment x e due to spring and frictional displacement x p x = x e + x p . The value of the stress acting on the spring is = E e = E ( p ) = E ✓ u s p ◆ Let the non negative variable : [0 , T ] ! denoted the internal hardening variable.We assume that we have the two following evolutionary equations˙ = | ˙ p | , and ˙ p = sign( ) , where > . (2.1)Then a yield criterion f ( , ), as a convex function, is defined f ( , ) := | | ( Y + K ) . (2.2)Observe that ˙ p = f ( , ) can be related to convex analysis, see Moreau [1973]. Thecriterion (2.5) allows to constraint the variables , to stay in the convex elastoplasticdomain , defined as follows:= ( , ) s.t. | | ( Y + K ) . (2.3)2 Let a 1D model (see Fig 2.1) composed of an elastic spring of length at rest and withYoung modulus E . At one extremity of the spring we fixe a mass m , and on the otherside we fixe a frictional device characterize by its cone of friction, that is the stress maytake any value from zero up to Y . and introduce the fundamental concept of return mapping or catching up algo-rithm. As shown in Chapter this notion has a straightforward generalization tothree-dimensional models and constitutes the single most important concept incomputational plasticity. In Section we illustrate the role of these integrativealgorithms by considering the simplest finite-element formulation of the elasto-plastic boundary-value problem. We discuss the incremental form of this problemand introduce the important notion of consistent or algorithmic tangent modulus .Finally, Section generalizes the preceding ideas to accommodate rate-dependent response within the framework of classical viscoplasticity. We examinetwopossibleformulationsofthisclassofmodelsanddiscusstheirnumericalimple-mentation. In particular, emphasis is placed on the significance of viscoplasticityas a regularization of rate-independent plasticity. This interpretation is importantin the solution of boundary-value problems where hyperbolicity of the equations inthepresenceofsofteningcanalwaysbeattainedbysuitablechoiceoftherelaxationtime.For further reading on the physical background, and generalizations, seeLemaitre and Chaboche [1990]. To motivate the mathematical structure of classical rate-independent plasticity,developed in subsequent sections, we examine the mechanical response of theone-dimensional frictional device illustrated in Figure .We assume that the device initially possesses unit length (and unit area) andconsists of a spring, with elastic constant E , and a Coulomb friction element, withconstant σ Y >
0, arranged as shown in Figure . We let σ be the applied stress(force) and ε the total strain (change in length) in the device. Inspection of Figure leads immediately to the following observations: ! ! Y ! E Figure 1.1.
One-dimensional frictional device illustrating rate–independent plasticity.
Figure 2.1:
Elastoplastic bar.
We admit an additive decomposition of the total displacement x into elastic displace-ment x e due to spring and frictional displacement x p x = x e + x p . The value of the stress acting on the spring is = E e = E ( p ) = E ✓ u s p ◆ Let the non negative variable : [0 , T ] ! denoted the internal hardening variable.We assume that we have the two following evolutionary equations˙ = | ˙ p | , and ˙ p = sign( ) , where > . (2.1)Then a yield criterion f ( , ), as a convex function, is defined f ( , ) := | | ( Y + K ) . (2.2)Observe that ˙ p = f ( , ) can be related to convex analysis, see Moreau [1973]. Thecriterion (2.5) allows to constraint the variables , to stay in the convex elastoplasticdomain , defined as follows:= ( , ) s.t. | | ( Y + K ) . (2.3)2 Let a 1D model (see Fig 2.1) composed of an elastic spring of length at rest and withYoung modulus E . At one extremity of the spring we fixe a mass m , and on the otherside we fixe a frictional device characterize by its cone of friction, that is the stress maytake any value from zero up to Y . and introduce the fundamental concept of return mapping or catching up algo-rithm. As shown in Chapter this notion has a straightforward generalization tothree-dimensional models and constitutes the single most important concept incomputational plasticity. In Section we illustrate the role of these integrativealgorithms by considering the simplest finite-element formulation of the elasto-plastic boundary-value problem. We discuss the incremental form of this problemand introduce the important notion of consistent or algorithmic tangent modulus .Finally, Section generalizes the preceding ideas to accommodate rate-dependent response within the framework of classical viscoplasticity. We examinetwopossibleformulationsofthisclassofmodelsanddiscusstheirnumericalimple-mentation. In particular, emphasis is placed on the significance of viscoplasticityas a regularization of rate-independent plasticity. This interpretation is importantinthesolutionofboundary-valueproblemswherehyperbolicityoftheequationsinthepresenceofsofteningcanalwaysbeattainedbysuitablechoiceoftherelaxationtime.For further reading on the physical background, and generalizations, seeLemaitre and Chaboche [1990]. To motivate the mathematical structure of classical rate-independent plasticity,developed in subsequent sections, we examine the mechanical response of theone-dimensional frictional device illustrated in Figure .We assume that the device initially possesses unit length (and unit area) andconsists of a spring, with elastic constant E , and a Coulomb friction element, withconstant σ Y >
0, arranged as shown in Figure . We let σ be the applied stress(force) and ε the total strain (change in length) in the device. Inspection of Figure leads immediately to the following observations: ! ! Y ! E Figure 1.1.
One-dimensional frictional device illustrating rate–independent plasticity.
Figure 2.1:
Elastoplastic bar.
We admit an additive decomposition of the total displacement x into elastic displace-ment x e due to spring and frictional displacement x p x = x e + x p . The value of the stress acting on the spring is F = Ex e = E ( x x p ) 0Then a yield criterion f ( , ), as a convex function, is defined f ( , ) := | | ( Y + K ) . (2.1)Observe that ˙ p = f ( , ) can be related to convex analysis, see Moreau [1973]. Thecriterion (2.4) allows to constraint the variables , to stay in the convex elastoplasticdomain , defined as follows:= ( , ) s.t. | | ( Y + K ) . (2.2)2 Figure 3.1:
Elastoplastic bar.
We admit that the total strain denoted ✏ is defined as the sum of the elasticstrain ✏ e and the plastic strain ✏ p , i.e., ✏ := ✏ e + ✏ p . (3.1)The total strain is measured as the change in length ` divided by the originallength ` . Concerning our model we choose ` = 1. This hypothesis allows toidentify the elongation of the spring with the elastic strain ✏ e . At initial time t we admit that the plastic strain ✏ p is 0. The decomposition (3.1) is valid whenplastic strain and elastic strain are small, see Lubliner [1990] p.486 and Maugin[2011] p.46. Which fits well with the rheological models.We deduce from our model that the stress due to the elastic strain and thestored energy associated are respectively given by = E ✏ e = E ( ✏ ✏ p ) and W ( ✏ ) = 12 E | ✏ ✏ p | . (3.2)With perfect plasticity, from Theorem 2.12, we can describe the di↵erent situ-ations that one meets:1. Outside of the plastic phenomenon, when v ✏ p = 0, the CEL equation describethe motion of the system. Whenever the CEL equations are satisfied, if thesystem is not dissipative (e.g., without damping term), the time energy-evolution equation is equal to zero.2. The plasticity phenomenon occurs at time t when f ( ) = 0 holds(a) the laws of plasticity (2.4) are verified, i.e., v ✏ p N ( ), Due to vertical jump conditions in (2.32) we get: For all open subsets U ⇢ U with piecewise smooth boundary and for all ⇠ in the Lie algebra g of the Lie group G , we have Z X ( D ) s @L@' A,µ ⇠ AY d n x µ { + Z @ X ( U ) \ X ( D ) ✓ @L@' A,µ ⇠ AY d n x µ ◆ = 0 , (2.36)where ⇠ Y is the infinitesimal generator. Remark 2.14
Generally nonsmoothness is associated with boundary contact,and friction, and/or with interior plasticity. In such cases the question of conser-vation of symmetries must be studied under combination of di↵erent perspectives. ⇤ Multisymplectic form formula.
Concerning the Cartan form (or multi-symplectic form) and the multisymplectic form formula this will be studied in itsown for various types of nonsmooth problems in a paper to come.
The rheological model we consider was described in Simo and Hughes [1998].
The 1D rheological model (see Fig 3.1) is composed of an elastic spring of length ` at rest, with Young modulus E , and elastic strain ✏ e . At one end of the springwe fix a mass m , and at the opposite we fix a frictional pad which induces thefrictional strain ✏ p referred to as the plastic strain. Figure 3.1:
Elastoplastic bar.
We admit that the total strain denoted ✏ is defined as the sum of the elasticstrain ✏ e and the plastic strain ✏ p , i.e., ✏ := ✏ e + ✏ p . (3.1)The elastic strain is measured as the change in length ` divided by the originallength ` . Concerning our model we choose ` = 1. This hypothesis allows toidentify the elongation of the spring with the elastic strain ✏ e . At initial time t we admit that the plastic strain ✏ p is 0. The decomposition (3.1) is valid whenplastic strain and elastic strain are small, see Lubliner [1990] p.486 and Maugin[2011] p.46. Which fits well with the rheological models. Figure 3.1:
Elastoplastic bar.
We admit that the total strain denoted (cid:15) is defined as the sum of the elasticstrain (cid:15) e and the plastic strain (cid:15) p , i.e., (cid:15) := (cid:15) e + (cid:15) p . (3.1)The elastic strain is measured as the change in length ∆ (cid:96) divided by the originallength (cid:96) . Concerning our model we choose (cid:96) = 1. This hypothesis allows toidentify the elongation of the spring with the elastic strain (cid:15) e . At initial time t we admit that the plastic strain (cid:15) p is 0. The decomposition (3.1) is valid whenplastic strain and elastic strain are small, see Lubliner [1990] p.486 and Maugin[2011] p.46. Which fits well with the rheological models.We deduce from our model that the stress due to the elastic strain and thestored energy associated are respectively given by σ = E (cid:15) e = E ( (cid:15) − (cid:15) p ) and W ( (cid:15) ) = 12 E | (cid:15) − (cid:15) p | . (3.2) The frictional device is characterized by the yield criterion f which constrains theadmissible stress σ to lie in the admissible set (2.3).The Lagrangian L ( (cid:15) , ˙ (cid:15) ) associated to perfect plasticity is defined by L ( (cid:15) , ˙ (cid:15) ) = 12 m | ˙ (cid:15) | − E | (cid:15) e | = 12 m | ˙ (cid:15) | − E | (cid:15) − (cid:15) p | . (3.3) ue to the plasticity laws described in § (cid:15) = ˙ (cid:15) e + (cid:74) (cid:15) p (cid:75) t , where (cid:74) (cid:15) p (cid:75) t = 0 when x / ∈ D X , i.e., (cid:15) p = Cte apart fromplastic domain.From Theorem 2.12, we can describe the different situations that one meets:
1. Elastic regime.
The elastic regime prevails as long as (cid:74) (cid:15) p (cid:75) t = 0. It isequivalent to saying that (cid:15) p = constant. The unconstrained CEL equations (3.5)describe the motion of the system. Whenever the CEL equations are satisfied, thetime energy-evolution equation is equal to zero. On the time interval [0 , T ] theaction map to be S ns ( (cid:15) ) = (cid:90) T L ( (cid:15) , ˙ (cid:15) ) dt. (3.4)Computing the variation of the action map d S ns ( (cid:15) ) · δ (cid:15) we get, from the Hamiltonprinciple, the Euler-Lagrange equations m ¨ (cid:15) + E ( (cid:15) − (cid:15) p ) = 0 . (3.5)In addition from the horizontal variations δt we get the conservation of energy.
2. Plastic regime.
The plastic regime occurs at time t when f ( σ ) = 0 holds1. the laws of plasticity (2.5) are verified, i.e., (cid:74) (cid:15) p (cid:75) t ∈ N E σ ( σ ),2. From Theorem 2.12 it follows that(a) the vertical jump condition induces momenta conservation ∂K ( ˙ (cid:15) ) /∂ ˙ (cid:15) due to the absence of constraints on (cid:15) ; see (3.6).(b) the horizontal energy jump (cid:74) − E tot (cid:75) is exactly equal to the plastic dissi-pation (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) ; see (3.8).From the vertical jump condition (2.32) we get (cid:74) m ˙ (cid:15) (cid:75) = 0 . (3.6)Due to plastic laws the rate of change of plastic strain satisfies (2.5), i.e., (cid:74) (cid:15) p (cid:75) t ∈ N E σ ( σ ) . (3.7)At time t the horizontal jump conditions (2.33) give the rate of change of thetotal energy, i.e., there exists λ such that (cid:74) − E tot (cid:75) = (cid:104) σ , λ∂ σ f ( σ ) (cid:105) = D p ( σ , λ∂ σ f ( σ )) , (3.8)where (cid:74) (cid:15) p (cid:75) ¯ t = λ∂ σ f ( σ ) and E tot ( (cid:15) , ˙ (cid:15) ) = 12 m | ˙ (cid:15) | + 12 E | (cid:15) − (cid:15) p | . (3.9) Example 3.1
Let the Tresca criterion defined in Tresca [1872] as follows f ( σ ) := | σ | − σ Y (cid:54) , with σ Y > . (3.10) We deduce the expression of the normal cone to E σ at σ N E σ ( σ ) = λ ∂f ( σ ) = λ sgn( σ ) , for all λ (cid:62) . he plastic phenomenon occurs when f ( σ ) = 0 , i.e., when | σ | = σ Y . We get (cid:74) (cid:15) p (cid:75) ¯ t = λ sgn( σ ) with λ > and D p ( σ , (cid:74) (cid:15) p (cid:75) ¯ t ) := λ | σ | . (3.11) The implementation of this example and the next two in § § Figure 3.2:
Tresca criterion.
From left to right: strain ( (cid:15) , (cid:15) p ), total energy, stress, and stress/strain. E = 30, m = 0 . In Figure 3.2 observe that the plastic strain (cid:15) p stops to evolve when | σ | < σ Y .Then after we go back to the elastic behavior and energy conservation. ♦ Internal strain hardening variables ξ = ( ξ i , ξ k ) are often added to the plasticstrain (cid:15) p , where ξ i , ξ k are respectively the isotropic and kinematic strain hardeningvariables. Then the potential energy is seen as the sum of the elastic store energyfunction (3.2) plus the potential function H ( ξ ) for the hardening variables.The yield criterion associated to pure plasticity (2.3) can be modified in twoways. a) Isotropic hardening:
The yield surface expands with increasing stress.Such that the yield criterion on ( σ , β i ) is now defined as f ( σ , β i ) := F ( σ ) − k σ Y ( β i ) (cid:54) β i = − ∂ H ( ξ ) /∂ξ i . (3.12) b) Kinematic hardening:
The yield surface with the same shape is translated instress space, with the following yield criterion f ( σ , β k ) := F ( σ , β k ) − σ Y (cid:54) β k = − ∂ H ( ξ ) /∂ξ k . (3.13)Note that the isotropic and kinematic hardening are often combined. But inthe following we will consider the two cases separately. We consider now the possibility of an expansion of the yield surface due to theincreasing flow stress k σ Y ( β i ). See, e.g., in Simo [1998] the following elementarymodel k σ Y ( β i ) = σ Y − β i where σ Y > E ( σ,q ) := { ( σ , β i ) ∈ R | f ( σ , β i ) ≤ } , (3.14) here the yield criterion f ( σ , β i ) for isotropic hardening verifies (3.12). The La-grangian is now defined by L ( (cid:15) , ˙ (cid:15) ) = 12 m | ˙ (cid:15) | − E | (cid:15) − (cid:15) p | − H ( ξ i ) . (3.15)Through the derivative of the action S ns ( (cid:15) ) outside of plastic behavior we get theEuler-Lagrange equations (3.5).When the plastic phenomenon occurs at time t from the vertical jump conditionwe get the same as the one obtained with perfect plasticity, i.e., (3.6). While fromthe plasticity law (2.5) we obtain the following rate of change of plastic strain andof isotropic hardening (cid:0) (cid:74) (cid:15) p (cid:75) ¯ t , (cid:74) ξ i (cid:75) ¯ t (cid:1) ∈ N E ( σ,q ) ( σ , β i ) = λ ∇ ( σ ,β i ) f ( σ , β i ) , for all λ ≥ . (3.16)The rate of change of the total energy (3.8) becomes (cid:74) − E tot (cid:75) = (cid:10) ( σ , β i ) , λ i ∇ ( σ ,β i ) f ( σ , β i ) (cid:11) =: D pih , with λ i > , (3.17)where D pih is denoted the isotropic hardening plastic dissipation function . Example 3.2
Let the 1D yield criterion corresponding to isotropic hardening f ( σ , β i ) := | σ | + β i − σ Y ≤ . (3.18) where σ Y is constant. We specify the potential function for isotropic hardeningvariables H ( ξ i ) = 12 Kξ i with K > . So we get (cid:20) σ β i (cid:21) = (cid:20) E ( (cid:15) − (cid:15) p ) − Kξ i (cid:21) and (cid:20) (cid:74) (cid:15) p (cid:75) ¯ t (cid:74) ξ i (cid:75) ¯ t (cid:21) ∈ N E ( σ,q ) ( σ , β i ) = λ (cid:20) sgn( σ )1 (cid:21) , for all λ ≥ . The numerical tests implemented through a discrete formulation of isotropichardening give us the following results
Figure 3.3:
From left to right: strain ( (cid:15) , (cid:15) p ), total energy, stress, and stress/strain. E = 30, K = 50, m = 0 . When the stress satisfies | σ | = σ Y − β i , the plastic strain increases throughsmall jumps, and stops as soon as | σ | < σ Y − β i . However note that the yieldsurface expands, due to σ Y − β i which increases. That is, loading after unloadingwill define a new instantaneous elastic limit and so forth. ♦ .3.2 kinematic hardening The yield criterion f ( σ , β k ) described by (3.13) exhibits kinematic hardening . See,e.g., in Simo [1998] the following elementary model F ( σ , β k ) = | σ − β k | . Wherethe yield surface translates in the direction of the plastic flow.When the plastic phenomenon occurs at time t , the rate of change of plasticstrain and kinematic hardening are given by( (cid:74) (cid:15) p (cid:75) ¯ t , (cid:74) ξ k (cid:75) ¯ t ) ∈ N E ( σ,q ) ( σ , β k ) = λ ∇ ( σ ,β k ) f ( σ , β k ) , for all λ ≥ . (3.19)While the horizontal jump condition becomes (cid:74) − E tot (cid:75) = (cid:10) ( σ , β k ) , λ k ∇ ( σ ,β k ) f ( σ , β k ) (cid:11) =: D pkh , with λ k > , (3.20)where the kinematic hardening plastic dissipation function is denoted by D pkh . Example 3.3
Let the 1D yield criterion corresponding to kinematic hardening f ( σ , β k ) := | σ − β k | − σ Y ≤ , (3.21) and the following potential function H ( ξ k ) = 12 H ( ξ k ) with H > . We get (cid:20) σ β k (cid:21) = (cid:20) E ( (cid:15) − (cid:15) p ) − Hξ k (cid:21) and (cid:20) (cid:74) (cid:15) p (cid:75) ¯ t (cid:74) ξ k (cid:75) ¯ t (cid:21) ∈ N E ( σ,βk ) ( σ , β k ) = λ (cid:20) sgn( σ − β k ) − sgn( σ − β k ) (cid:21) , ∀ λ ≥ . The numerical tests implemented through a discrete formulation of kinematichardening give us the following results
Figure 3.4:
From left to right: strain ( (cid:15) , (cid:15) p ), total energy, stress, and stress/strain. E = 30, H = 35, m = 0 . Note that the yield surface retains the same shape but translates during theplastic strain, due to the internal variable β k in | σ − β k | . ♦ “First, suppose that we do irreversible work on an object by friction, generating aheat Q on some object at temperature T . The entropy is increased by Q/T . Theheat Q is equal to the work W , and thus when we do a certain amount of work byfriction against an object whose temperature is T , the entropy of the whole worldincreases by W/T .”(Feynman, Leighton, and Sands [1963])The two main points which characterize the cristal plastic phenomenon are asfollows: first, the heat is produced by the plastic strains; second, in the irreversibleplastic change, the total entropy of the system always increases. .1 First and second laws of thermodynamics In the following we recall the second and first laws of thermodynamics.
Second law.
The second law of thermodynamics was first put into words byCarnot [1824]. It can be expressed as follows:In a isolated mechanical system which absorbes heat Q at temperature T anddelivers heat Q at temperature T , the relation between the two verifies Q T = S = Q T , (4.1)where S denotes the entropy. However the second law of thermodynamics is ex-pressed through different forms which depend from the perspective adopted. Forexample, the Clausius-Duhem local form of the “second law of thermodynamics”in a continuous body asserts that (see e.g., Marsden and Hughes [1994] § γ = ˙ S − ρrT + 1 T ∇ · q − T ∇ T · q ≥ , (4.2)where γ is the rate of change of entropy production, S is the entropy in the body, r is the heat supply by unit of mass, T is the temperature, q is the heat flux, and˙ S is the rate of change of the total entropy.The dissipation T γ is decomposed into the sum of the internal dissipation D int under the Clausius-Plank form of the second law and the dissipation D cond arisingfrom heat conduction, see Truesdell and Noll [1965](79.8, 79.9, 79.10), which arerespectively D int := T ˙ S − ρr + ∇ · q ≥ , and D cond := − T ∇ T · q ≥ . (4.3)In the expression (4.3) note that we take into account the heat flux q which isfunction of the thermal conductivity of the material. First law or balance of energy.
Clausius [1850] and W. Rankine bothstated the first law of thermodynamics which says that the rate of increase of theinternal energy ˙ E int of the body equals the rate of work done (the body forces andsurface traction) plus the rate of increase of heat energy.˙ E int = ˙ W + ˙ Q. (4.4)As a consequence, if we admit that the transformation is isothermal, then thework dissipated during plastic deformation is transformed into heat, i.e., the worklost by internal friction is equal to the heat produced.˙ Q plastic = − ˙ W friction at temperature T. We recall from Marsden and Hughes [1994] § E int = σ : d + ρr − ∇ · q , (4.5)where σ is the Cauchy stress tensor and d is the rate of change of strain tensor. y combination of the Clausius-Duhem inequality (4.3) and the energy bal-ance (4.5) we can express the internal dissipation D int in the solids as the thermalpower plus the mechanical power minus the time rate of change of the internalenergy, i.e., D int = T ˙ S + σ : d − ˙ E int ≥ . (4.6) Thermomechanics of plasticity.
We introduce temperature T as a newvariable in the 1D elastoplastic model described in section § W ( (cid:15) e , S e ). Then, we recall thefollowing local state axioms ; see e.g., Maugin [1992] σ = ∂ (cid:15) e W ( (cid:15) e , S e ) , T = ∂ S e W ( (cid:15) e , S e ) . (4.7)We admit that the total entropy S is the sum of the entropy S e due to theelasticity and the entropy S p due to the plasticity, i.e., S = S e + S p . (4.8)The Helmholtz free energy Ψ( (cid:15) e , T ) is defined from W by performing the changeof variable S e → T through the Legendre-Fenchel transform, see e.g. Ottinger[2005] Ψ( (cid:15) e , T ) = W ( (cid:15) e , S e ) − T S e . (4.9)Hence, the local state axioms (4.7) are now expressed as follows σ = ∂ (cid:15) e Ψ( (cid:15) e , T ) , S e = − ∂ T Ψ( (cid:15) e , T ) . (4.10) Thermoelastic regime.
The mechanical system is described by its Lagrangiancomposed of the kinetic energy minus the Helmholtz free energy L T ( (cid:15) , ˙ (cid:15) ) = 12 m | ˙ (cid:15) | − Ψ( (cid:15) − (cid:15) p , T ) . (4.11)Through the derivative of the action S ns ( (cid:15) ) = (cid:82) L T ( (cid:15) , ˙ (cid:15) ) with respect to (cid:15) we getthe Euler-Lagrange equation m ¨ (cid:15) + ∂ (cid:15) Ψ( (cid:15) − (cid:15) p , T ) = 0 . (4.12) Remark 4.1
Note that we could define the Lagrangian (4.11) by taking intoaccount the internal energy W ( (cid:15) e , S e ) instead of the free energy (4.9), with asimilar result. (cid:3) Internal dissipation.
The general constitutive equations (4.6) provides theinternal plastic dissipation D int , i.e., D int = T ( ˙ S e + (cid:74) S p (cid:75) t ) + (cid:10) σ , ˙ (cid:15) e + (cid:74) (cid:15) p (cid:75) t (cid:11) − ˙ W ( (cid:15) e , S e ) (4.7) = T (cid:74) S p (cid:75) t + (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) ≥ , (4.13) here the plastic strain (cid:74) (cid:15) p (cid:75) t evolves by jumps, and therefore the rate of changeof plastic entropy (cid:74) S p (cid:75) t evolves also by jumps.We deduce the decomposition of the internal dissipation (4.13) into mechanicaldissipation D mech and thermic dissipation D ther , which are respectively D mech ( σ , (cid:74) (cid:15) p (cid:75) t ) := (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) and D ther ( T, (cid:74) S p (cid:75) t ) := T (cid:74) S p (cid:75) t . (4.14) Thermoelastoplastic domain.
In the context of thermoplasticity the in-ternal plastic dissipation (4.13) verifies the maximum dissipation principle; seeLubliner [1984], Simo [1998] § T . Therefore, the thermoelastoplastic domain is defined as follows E σ,T := { ( σ , T ) | f ( σ , T ) ≤ } , (4.15)where Int( E σ,T ) and ∂ E σ,T define respectively the thermoelastic and the thermo-plastic domains.Given the maximum dissipation principle the problem we have to solve is tominimize −D mech and −D ther under the constraint f ( σ , T ) ≤
0. From the resultsrecalled in § (cid:20) (cid:74) (cid:15) p (cid:75) t (cid:74) S p (cid:75) t (cid:21) ∈ N E σ,T ( σ , T ) = λ (cid:20) ∂ σ f ( σ , T ) ∂ T f ( σ , T ) (cid:21) , for all λ ≥ , or equivalently (cid:20) σ T (cid:21) ∈ N ∗ E σ,T ( (cid:74) (cid:15) p (cid:75) t , (cid:74) S p (cid:75) t ) . (4.16)From (4.16) we get ( (cid:74) (cid:15) p (cid:75) t , (cid:74) S p (cid:75) t ). Elastic entropy.
The expression (4.13) issued from (4.6) together with theconstitutive equations (4.3) yields the following relation T ˙ S e = (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) − ∇ · q . (4.17)By assumption we admit that there is no heat flux nor heat supply in the 1Delastoplastic model described in section § ∇· q = 0. So we get the followingrate of change of elastic entropy T ˙ S e = (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) ⇔ ˙ S e = 1 T (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) . (4.18)The mechanical dissipation (4.14) is described by jumps of energy, see § S e evolve by jumps. From now on the rate of change of elasticentropy is denoted (cid:74) S e (cid:75) t := 1 T (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) . (4.19)In a logical way, outside of plastic phenomenon, the rate of change of elasticentropy (4.19) becomes (cid:74) S e (cid:75) t = 0. otal energy and entropy production. The total energy is composed ofthe kinetic energy m | ˙ (cid:15) | , associated to the rheological model defined in § W ( (cid:15) e , S e ). The rate of change of the total energy (cid:74) E tot (cid:75) t , canbe derive from (4.13), i.e., (cid:74) E tot (cid:75) t (4.7) = ˙ (cid:15) m ¨ (cid:15) + (cid:104) σ , ˙ (cid:15) e (cid:105) + T (cid:74) S e (cid:75) t (3.1)(4.12) = − (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) + T (cid:74) S e (cid:75) t (4.19) = 0 , (4.20)where we used the relation ˙ (cid:15) = ˙ (cid:15) e + (cid:74) (cid:15) p (cid:75) t . Hence the total energy is conserved.Finally the rate of change of the entropy production (4.2) has the following ex-pression γ = (cid:74) S e (cid:75) t + (cid:74) S p (cid:75) t = 1 T (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) + (cid:74) S p (cid:75) t (4.16) = λT (cid:104) σ , ∂ σ f ( σ , T ) (cid:105) + λ ∂ T f ( σ , T ) , (4.21)where γ evolves by jumps. Example 4.2 : Let the following yield criterion issued from Tresca criterion (3.10) f ( σ , T ) = | σ | − σ Y ( T ) (cid:54) , During the plastic phenomena, when f ( σ , T ) = 0 , we get the mechanical dissi-pation and the thermic dissipation from (4.16) : D mech ( σ , (cid:74) (cid:15) p (cid:75) t ) = λ | σ | , and D ther = − λT ddT σ Y ( T ) . In addition we obtain γ = λT | σ | − λ ddT σ Y ( T ) ≥ . ♦ Let the rheological model with isotropichardening law, as described in § W ih ( (cid:15) e , ξ i , S e ) = W ( (cid:15) e , S e ) + H ( ξ i ) , (4.22)whereas the relationships (4.7) are transformed into σ = ∂ (cid:15) e W ih ( (cid:15) e , ξ i , S e ) , β i = − ∂ ξ i W ih ( (cid:15) e , ξ i , S e ) , T = ∂ S e W ih ( (cid:15) e , ξ i , S e ) . (4.23)Then, the Helmotz free energy is now defined as followsΨ ih ( (cid:15) e , ξ i , T ) = W ih ( (cid:15) e , ξ i , S e ) − T S e , with (4.24) σ = ∂ (cid:15) e Ψ ih ( (cid:15) e , ξ i , T ) , β i = − ∂ ξ i Ψ ih ( (cid:15) e , ξ i , T ) , S e = − ∂ T Ψ ih ( (cid:15) e , ξ i , T ) . (4.25) nternal dissipation. For the isotropic hardening law we get the internaldissipation D int from the general constitutive equations (4.6) and (4.23), i.e., D int = (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) + (cid:104) β i , (cid:74) ξ i (cid:75) (cid:105) + T (cid:74) S p (cid:75) t = D mech + D ther ≥ , (4.26)which verify the maximum dissipation principle. The expression of the rate ofchange of the elastic entropy is obtained by (4.26) together with the constitutiveequations (4.3) . We get (cid:74) S e (cid:75) t = 1 T (cid:0) (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) + (cid:104) β i , (cid:74) ξ i (cid:75) t (cid:105) (cid:1) . (4.27) Thermoelastoplastic domain.
It is now defined as follows E ih := { ( σ , β i , T ) | f ih ( σ , β i , T ) ≤ } . (4.28)The problem to solve becomes: to minimize −D mech and −D ther under the con-straint f ih ( σ , β i , T ) ≤
0. We obtain (cid:74) (cid:15) p (cid:75) t (cid:74) ξ i (cid:75) t (cid:74) S p (cid:75) t ∈ N E h ( σ , β i , T ) = λ ∂ σ f ih ( σ , β i , T ) ∂ β i f ih ( σ , β i , T ) ∂ T f ih ( σ , β i , T ) , for all λ ≥ , or equivalently (cid:2) σ β i T (cid:3) T ∈ N ∗ E h ( (cid:74) (cid:15) p (cid:75) t , (cid:74) ξ i (cid:75) t , (cid:74) S p (cid:75) t ) . (4.29) Variational formulation.
The Lagrangian composed of the kinetic energyminus the free energy L T ( (cid:15) , ˙ (cid:15) ) = 12 m | ˙ (cid:15) | − Ψ ih ( (cid:15) − (cid:15) p , ξ i , T ) . (4.30)The CEL equation is obtained through the derivative of the action with respectof (cid:15) . The resulting expression is m ¨ (cid:15) + ∂ (cid:15) Ψ ih ( (cid:15) − (cid:15) p , ξ i , T ) = 0 . (4.31) Entropy production.
At temperature T the total energy is conserved andthe rate of change of entropy production γ becomes γ = (cid:74) S e (cid:75) t + (cid:74) S p (cid:75) t = 1 T (cid:16) (cid:10) σ , (cid:74) (cid:15) p (cid:75) t (cid:11) + (cid:10) β i , (cid:74) ξ i (cid:75) t (cid:11) (cid:17) + (cid:74) S p (cid:75) t (4.29) = λT (cid:16) (cid:10) σ , ∂ σ f ih ( σ , β i , T ) (cid:11) + (cid:10) β i , ∂ β i f ih ( σ , β i , T ) (cid:11) (cid:17) + λ ∂ T f ih ( σ , β i , T ) . (4.32) For the thermo kinematic hardening law we get the same expressions than in § β i and ξ i by β k and ξ k , associated witha new constraint f kh ( σ , β k , T ) ≤ H ( ξ k ) for thekinematic hardening variables. .4 Summary We summarise the previous results in the following proposition
Proposition 4.3
Let the 1D thermoelastoplastic model as described previously.Let the total strain (cid:15) and the total entropy S which are seen as the sum of theirelastic and plastic part, i.e., (cid:15) = (cid:15) e + (cid:15) p , S = S e + S p . (4.33) With the isotropic and kinematic strain hardening variables ξ = ( ξ i , ξ k ) , the tem-perature T , and the Helmhotz free energy Ψ h ( (cid:15) e , ξ , T ) . Given the yield criterion f h ( σ , β , T ) ≤ which constraint the stress tensor field σ = ∂ (cid:15) e Ψ h ( (cid:15) e , ξ , T ) , thestress hardening variables β = − ∂ ξ Ψ h ( (cid:15) e , ξ , T ) , and the temperature T to lie inthe thermoelastoplastic domain. If we assume that there are no heat flux nor heatsupply in our model. Then, at fixed temperature T , when plastic phenomenon oc-curs at time t the elastic and plastic entropy evolve by jumps, and their time rateof change are given by (cid:74) S e (cid:75) t = λT (cid:16) (cid:10) σ , ∂ σ f h ( σ , β , T ) (cid:11) + (cid:10) β , ∂ β f h ( σ , β , T ) (cid:11) (cid:17) , (cid:74) S p (cid:75) t = λ ∂ T f h ( σ , β , T ) , with λ > . (4.34) Remark 4.4
We recall that: “For every admissible process in a perfect material,the entropy production is zero” (Truesdell and Noll [1965]). In our case the entropyproduction is only due to plastic phenomenon. (cid:3)
As highlighted in this development, the elastoplasticity is a nonsmooth phenomenondescribed by a succession of dissipation jumps which interrupt the smooth path ac-counted for by a multisymplectic variational formulation. By opposition with vis-coelastoplastic dissipative phenomena which are smooth and not described througha variational formulation.Hence, the next important task is to develop discrete mechanics for nonsmoothelastoplasticity by taking advantage of the variational integrators (such as Fetecau,Marsden, Ortiz, and West [2003] and Demoures et al. [2017]) that are developingin that direction. This task is presently in progress in Demoures [2018c].There are several other directions to pursue. The most important is to in-clude friction in the nonsmooth multisymplectic variational formalism, which is adissipative phenomenon defined through a maximal principle in the same way aselastoplasticity.Then we need to associate different nonsmooth problems, like contact withplasticity, or friction with plasticity, or even contact, plasticity and friction thatrequire further attention in order to get a clear picture of these associations.
Acknowledgment.
I thank Doc. F. Gay-Balmaz for having welcomed meduring 6 months at LMD/IPSL, CNRS, Ecole Normale Sup´erieure Paris. MoreoverI wish to thank Prof. A. Curnier (EPFL) for many helpful discussions. eferences Armero, F. [2008]
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